Special powers that, somehow, rely on the power of traced shapes and diagrams to function. Almost always a circular design, with additional shapes of varying complexity inside the circle. Sometimes only required for a special ceremony, sometimes necessary for anything and everything remotely magical. Only very rarely will the mechanism for how the symbols affect the magic be explained.

A form of Hermetic Magic. Geometric Magic usually sets itself up as a type of Functional Magic, frequently paired with Ritual Magic. When the magic seems to spawn the shapes rather than the other way around, it's Instant Runes. Instant Runes are generally involved anyway as the activation of the shape may cause writing to appear.

Examples:

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Anime & Manga

Fullmetal Alchemist: Alchemy requires "transmutation circles" that are essentially Geometric Magic. The relationship between the various transmutation circles is never explained, though it's implied that more complicated designs are required for more complicated alchemy (a very basic transmutation circle consists of nothing more than a square inside a circle. Some of the symbols are actual alchemical symbols used way back when people were experimenting with various metals, chemicals, etc, and used the symbols to keep track of what they were doing). They're called "Arrays" and most Alchemists have to prepare them beforehand on items they possess or their bodies; e.g. Roy Mustang's gloves, Scar's tattoos. However these things are all drawn ridiculously fast, with the most egregious example being drawing half of one on two tables, and shoving them together, each taking about half a second.

Averted with philosopher's stones, which as a secondary feature allow the user to transmute things without circles. that's why people using (or "being") a philosopher's stone can transmute without a circle.

Ed, Izumi, and Al don't have to draw the circle, but the series explains that they are still forming an array: their clapped hands form a circle out of the body/arms with a simple triangle on one end. The rest of the details of the array are held in the alchemist's mind. You need immense knowledge of alchemy to pull this off; this is why the only people who can do it are ones who have opened the Gate. Consider it a perk of being crippled from attempting human transmutation.

In the manga this is why the Big Bad has to spend generations having a country-sized Philosopher's Stone array dug around the country in order to execute his plan.

Ga-Rei - A circular seal is used to hold Byakuei in check, and appears when the seal is partially released.

CLAMP's Cardcaptor Sakura 's title character formed a circle with astrological signs in it when she performed magic with the Clow cards and wand, (and later a different circle when she formed the Sakura cards). Then, in the opening chapters of Tsubasa Reservoir Chronicle, we see several different forms of magic perform the same function - sending a person or persons to the Witch of Dimensions - each with their own special magical designs.

Related: The Instant Runes in Magical Girl Lyrical Nanoha adopt different geometric shapes dependent on the class of magic the caster is using. There are at least 4 known variations, 2 of them corresponding to major schools of magic and one to magical specialisation.

The pactio spell in Mahou Sensei Negima! requires a magic circle to operate properly. Other spells also require circles to be drawn (or created out of Instant Runes), but as a mage grows more experienced with that spell, they lose the need to draw the circle. Generally the more complex and powerful the spell, the more complex the circle required to activate it.

Mahoujin Guru Guru: Kukuri's Guru Guru magic requires her to trace magic circles along the ground around her to cast spells and summon.

In "Howl's Moving Castle" Howl draws a magical diagram on the floor in chalk when preparing to change one of the castle real-world endpoints to become the house next to the hat shop.

Fan Fiction

In Discworld fic Whyand Were, Mustrum Ridcully demonstrates this to Ponder Stibbons and a sceptical Matron Igorina using nothing more than a length of string and a piece of chalk. An entire hospital bed plus occupant has to be lifted inside a very big magical octogram; the reasoning is that Ruth N'Kweze would go inside the circle, but the Howondalandian magical entity possessing her could not enter and she would therefore be freed.

Film

Ghostbusters even has a reference to this; Dana's apartment building was designed by a group of cultists in the 1930s with the specific purpose of serving as a "spiritual array" for the summoning of Gozer.

The Lament Configuration of Clive Barker's Hellraiser series and adaptations had variations in puzzle boxes and even whole buildings. Any of them could open a gate to Leviathan (hell) or summon, banish and control cenobites (demons) by finding the correct geometric state in the puzzle.

The Skeleton Key deals with Hoodoo in Louisiana, and some magic circles are seen, including during the climax where the protagonist thinks she's drawing a circle of protection around herself, but in fact traps herself inside the circle as the person who planted the instructions on her planned

Literature

The Dresden Files uses circles a lot, occasionally with other figures (like the Seal of Solomon, a pentagram, or a triangle) in it. Humans, however, aren't the only ones who can use this. The Fair Folk can use them as well. The key difference is a magic circle, any magic, circle crafted in the human realm and contains a native born human within, can summon something into the circle.

The protagonist prefers five-pointed stars, but states that he's used to amateurs using triangles because they're easier to draw.

Interestingly enough, Harry himself points out that technically, you don't need to actually make the closed geometric shape to work your magic, since it's all based on intent and willpower. If a person believes starting the pentagram from the top point gives it more power or more suited to attack magic, the end result will reflect this as well. Most everyone uses circles, triangles, etc anyway, because while you technically don't need them, they make it a hell of a lot easier to use the spell in question, and there's no reason not to when you're not using off-the-cuff evocations and the like.

And in Small Favor, Harry shows what happens when you use a deliberately imbalanced geometric model (e.g., a pentagram with one point longer than the others) for an invocation — a lot of energy let loose in an uncontrolled fashion. However, since the casters in question were effectively using it for a hellfire explosion...

Circles also have another advantage: they're quick. Magical fog swooping down on you? Take two salt shakers, spin, add a bit of willpower, and bam, you're safe. They're so quick and easy to use that any human who is given a bit of instruction can pull it off, regardless of ability with magic (although it costs a small blood sacrifice — a few drops is generally plenty — if they don't have some decent magical training).

In Cold Days, it turns out that the original Merlin was the grandmaster of this trope. Through application of this on an epic transtemporal and transdimensional scale, outright defying the laws of physics, he designs a prison in which Eldritch Abomination is minimum security.

The Bartimaeus Trilogy is the epitome of this trope. In this series of books, almost all magic is the result of summoning and confining spirits inside pentacles, which are described as being composed mainly of words. This is a necessary step because the aforementioned spirits are usually quite vengeful for their long imprisonments in the physical realm and most of them have developed a taste for human flesh.

In Shaman of the Undead, a circle is sometimes used by the shaman to talk with the ghost, although it's not necessary. On the other hand, necromancers and demon-binders need circles to do their jobs.

Pops up in the works of HP Lovecraft (there's the infamous Elder Sign, and The Dreams in the Witch-House goes so far as to say that magic is, in fact, mathematics) and his colleagues and imitators (Frank Belknap Long's The Hounds of Tindalos).

Related to the above, in The Laundry Series by Charles Stross, magic is an application of higher mathematics onto reality via dimension-bending "Dho-Na Curves". It's all very complicated, but suffice it to say that most magical rituals require a sort of "summoning grid", a geometric structure paired with some form of computation.

In The Mortal Instruments, this is the primary form of magic employed by the Shadowhunters, who use Angelic Runes to perform a variety of functions, including giving themselves superhuman physical abilities. Clary's special aptitude with Runes is essential to the plot.

Described vaguely in Garth Nix's Old Kingdom trilogy, 'Charter Magic' is more or less defined by the use of names and runic symbols ("Charter Marks"), each of which has a unique power. Charter Mages are 'baptized' with a Mark on their forehead soon after birth (usually).

All magic in The Wheel of Time is heavily geometric, involving "weaving threads" of the five elements into intricate patterns. Disturbing the pattern by trying to pick apart the weave (extremely difficult) and losing control (extremely easy) collapses the weave with unpredictable (but probably dangerous) results.

The Sword of Truth, contains several examples. The most powerful spells in the book require special signs drawn in powdered herbs, bone, blood, and, for the most powerful spells, sorcerer's sand (crystallized wizard bones from 3000 years ago). It is stated that in the last case, even the smallest mistake cannot be corrected and means death. At least one wizard family lived in an entire palace built as a magical sign. There is also a form of magic where an artist uses various shapes, formulas, and such in conjunction with a painting of a person in order to place curses on them.

Various spells and rituals in Discworld require octograms to be drawn on the floor. An example is the Rite of Ashk-Ente, which supposedly binds Death to the inside of the octogram... but he just stays in to be polite. Also, it is explained that most of the magical paraphernalia are used only for the look of the thing and are not really necessary, so it could be considered a subversion.

Apparently magic circles are actually of use, but don't require elaborate ritual: to prevent fallout from a magical duel Nanny Ogg uses the side of her foot to draw a big rough circle around them, confusing the apprentice witches who thought elaborate diagrams would be needed.

Circles are popular in the Evie Scelan novels, including I-95 surrounding all of Boston for one spell

Most wizard spells in the Young Wizards universe use this, excepting some magic that can be done solely with spoken Speech (the Masons Word, for example). Spell diagrams are likened to mathematical equations that must be balanced, simple ones can be 'drawn' in the caster's head, more complex ones are drawn out, pre-compiled in the wizard's manual or fashioned into illusion forms the wizard can place or carry inconspicuously, like Nita's charm bracelet, Kit's dimension-hopping leash for Ponch, etc.

In Good Omens the demon Crowley spends many years and much effort causing a highway to be built in the shape of an evil sigil, thus causing a malaise of bad vibes and mild evil to be produced by all the cars driving over the shape.

The motorway in question is the M25, which encircles London - anyone who has to make frequent use of it suspects this also to be the case in real life.

In Martha Wells' trilogy The Fall of Ile-Rien the wizards are busy reverse-engineering a kind of "travelling circle"-spell, where altering the symbols lets you go to different places and worlds.

Carol Berg's trilogy Collegia Magica features magic based on this principle.

The AonDor magic system in Elantris, which requires drawing symbols in the air with a lot of attention to detail and proportion. The geometry of the symbols is in turn based on the geography of the region; when a chasm was opened up by an earthquake, all the magic instantly stopped working until the symbols were updated to match.

Shows up again in another Sanderson story, The Emperor's Soul, where skilled forgers can carve magic stamps capable of changing anything they're used on. Word of God says it's set on the same world as Elantris, in The Verse known as The Cosmere.

A third example in the same world are the Dakhor monks, who inscribe mystic runes onto their bones to gain various powers, ranging from Super Strength to Anti-Magic.

Charles Stross's The Merchant Princes Series concerns a family with hereditary ability to jump between parallel worlds, triggered by looking at a Celtic Knot-style figure. (The protagonist learns about this the hard way when she examines a locket that belonged to her mother: she traces the twists and poof. As the figure becomes more familiar, it triggers her more quickly.) One member of the family made a flawed version of the figure from memory, and jumped into a different parallel world.

Later books in the series have the younger and more progressive Clan elements using the computer program Mathematica to determine what the two figures have in common, and what others can be drawn with the same properties.

The Laundry Series by the same author uses computers and highly advanced mathematics for summoning and binding demons for Functional Magic. (Okay, they're technically not demons, but they are creatures from alternate universes that tend to think human souls are crunchy and good with ketchup.)

The kanji in The Grimnoir Chronicles books are revealed to be 2D sympathetic representations of the various aspects that make up the Cosmic Entity from whom superpowers/magic spring.

The Witch Watch has sorcery that depends entirely on circles of writing. They can be written in various materials: chalk, grease pencil, blood, electrical wiring.

Ra goes in for REALLY BIG circles. Adjusting for the curvature of the earth big.

Spell-diars in Tough Magic are used to power and control spells. Although they're not absolutely neccessary, spellcasting being possible even without them, they still provide a significant boost in effiency.

Live Action TV

Used quite a bit on Supernatural. A wide variety of symbols and runes are used for protection against or to entrap different supernatural beings such as demons or angels. This is one of the few types of magic that Hunters regularly rely on.

Frequently used on America Unearthed, most notably and specifically in the episode Secret Blueprint of America but used frequently throughout the series.

Tabletop Games

Dungeons & Dragons: There is a class called the Geometer◊ that specifically uses Geometric Magic. This kind of magic is much more efficient for storing spells, among other abilities of the class.

Not to be confused with earlier editions' rules for certain spells (such as Lightning Bolt and Fireball) that actually required the players to use geometry to figure out the exact effect of the spell.

In Shadowrun, hermetic mages must spend several hours scribing hermetic circles for advanced magic such as sendings or summoning spiritual allies or elementals. Shamans, who are essentially the same but summon nature spirits, not elementals, do not need circles; instead, they rely on spirit lodges for their magic.

The Star Ocean series features this, where it's just called magic in the first game, but later it becomes Symbology (or Heraldry, in 3).

The former page image came from Pokémon HeartGold and SoulSilver; normally the series doesn't use this trope much/at all, but the image is from an event activated by Arceus, an event-giveaway Pokémon, who will create for the player an egg containing either the ruler of time, space, or antimatter.

Crops up once or twice in Super Robot Wars, most notably by Asakim Dorwin and his Shurouga in Z. He takes Goth tropes to absurd levels, having his Humongous Mecha cut its palm and trail "blood" (stay with me here) as he attacks the enemy from every angle. The screen then zooms out to show that his "blood trail" has formed a diagram, causing an explosion. Another attack has him create the same diagram, then fly through it in order to... um... "bust leys" and make gothic artwork appear.

Eternal Darkness: One of the few times that the mechanics of the system is explained — each spell is a combination of various glyphs that translate into specific words, essentially making each spell a sentence. Not quite Instant Runes; they take a few seconds and can be cancelled by the slightest movement of the caster.

Comes in triangles, pentacles, and septacles, by the way. The only difference is that the spell is stronger with more points; a fourth rune is added multiple times to the same basic three-rune spell to increase the power. At the end, it uses an anneagram (a nine-pointed star), which doesn't seem that much more impressive until you realize that the spell pattern is now the size of an entire city.

Although most of the symbols in Tales of the Abyss are of the Instant Runes variety, there are a few instances where more powerful magic is required, and huge fonic glyphs are drawn to accommodate their power. There's actually a brief mini-game where you have to re-draw one that you were meaning to use as an escape route (on a time limit!) because a villain wiped it out.

Mega Man Star Force: Solo always opens his Transformation Sequence by drawing the runic symbol of Mu in the air with his Ancient Star Carrier. Then a few more blink into existence, and suddenly everything goes to hell.

In Rhiannon: Curse of the Four Branches, Boswell's wand was hidden away using elemental sigils arranged in a pentagram. The elemental orbs, together with the chapel door (?), form another pentagram made of light to seal away the Big Bad.

One of Runescape's older quests, Legend's Quest, has a few uses of this trope. You are told to seek out a shaman in a cave which is marked by rocks in the the holy shape (a triangle). The shaman, as your character doesn't realize until after freeing him, has become possessed and used his last moments of sanity to lock himself in a flame barrier which is in the shape of some type of star.

Visual Novels

Juniper's Knot: The fiend is trapped within a magic circle and cannot step outside it without facing excruciating pain. The only way she can safely get out is if another life enters the circle to take her place.

Talies Heoria, from The Motley Two, is a WRITER-MAGE and thus his Weapon of Choice are pens, with which he draws symbols on sheets of paper, releasing magical effects. He has a tendency to scribble randomly, and sometimes accidentally ends up drawing an unwanted rune.

You spend the minutes after your conversation doodling idly in your sketchbook. Only one of them blows up in your face. Progress!

SCP Foundation: SCP-259 is, as someone in its discussion page noticed, essentially a modern-day version of a magical diagram: a fractal which, when rendered on a computer, opens a portal to Another Dimension.

Real Life

Truth in Television: Plenty of New Age gurus claim that the Pyramids of Egypt were built on cosmic geometries, which preserved the remains of the mummies inside and could do wonders in the modern day. Busted by the MythBusters, and one of the last "oogie-boogie" myths they did on the show.

Wiccans arrange their ritual spaces into specific patterns and use symbols of varying complexity in their religious rituals.

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